Indian Institute of Science Education and Research Thiruvananthapuram

IISER TVM Decennial Celebrations



2-d Navier Stokes equations: controllability, quasipotential and exit time asymptoics for small noise perturbations.

We prove that for every finite enstrophy divergence free vector field $\phi$ we can find a solution $u$ to the inhomogeneous 2-d Navier Stokes Equations over negative times such that $u(-\infty)=$ and $u(0)=0$ and such the corresponding time-dependent finite energy external force $f$. Given $\phi$, the infimum of the energy over all possible solutions is called the quasipotential $\mathbf{U}(\phi)$. We find an explicit expression for  \mathbf{U}(\phi)$ as well as the corresponding force in the periodic boundary conditions case. We also study the $\Gamma$ convergence of quasipotentials  \mathbf{U}_\delta(\phi)$ towards $\mathbf{U}(\phi)$, where $\mathbf{U}_\delta(\phi)$ is the infimum of the energies as above but calculated in a different way. Finally, we mention applications of this convergence for 2-d Navier Stokes Equations perturbed by small noise for two problems: asymptotics of the exit time and large deviations of the invariant measure. 

Date and Time: 2015/12/03 14:00 Venue: Seminar room (219)
Speaker: Prof: Zdzislaw Brzezniak, Speaker - Affiliations: York University, UK